There's lots of reasons why I don't play chess nowadays (how many games in ths database?) but I think it is truely a great game. I realise that there are many varients so I'd onnly like significant ones listed, but I'll leave that to you.
Designed to use the same board and pieces provided in a standard Chess set, Arimaa is the first game created intentionally to be difficult for computers to play. The number of possible moves at each turn in Arimaa is almost 1000 times that of Chess.
There is a reward of $10,000 for the first person or company that can build a computer program that can defeat a selected human Arimaa player prior to 2020.
The complete rules of the game are available at the http://www.arimaa.com/ website where visitors can also use an online tutorial.
Although I haven't played this, I stumbled upon it when I wasn't checking out current abstract gaming fare. It was a 2001 winner at some notable game design competition. Click the game's name for details.
As Sonja mentioned, a lot of Icehouse games use a chessboard. This is one of my favorites; it's a game where you build little programs of 5 instructions and then everyone's programs execute simultaneously.
Running Six on a Checkerboard...is a modification of the Kookaburra Game, and (no surprise) uses the beautifully sculpted Kookaburra Game pieces. RSoaC is only for 2 players; both players have an extra kangaroo; and the rules are modified as well, to make it an adult strategy game.
This one is a stretch for this list. Force Intriguing Games has a 24-space track (block off corner spaces on the chessboard), and players head for the center 4 squares for "home." First player to get 3 soldiers home wins.